In general, a succinct representation (also called circuit description in our setting) of a binary word \(w\)  is a boolean circuit that on input \(i\)  in binary emits two boolean values \(x,y\)  as an output: \(x=1\)  iff \(i\le |w|\) , and if \(x=1\)  then \(y\)  is the \(i\) 'th bit of \(w\)  [1]}.
In that sense the circuit description of an integer \(k\)  is a boolean circuit that on input \(i\)  in binary emits 1 iff \(i\le k\) , and the
circuit description of a binary relation \(R\)  of natural numbers is a boolean circuit \(C_R\)  that on input \((i,j)\)  in binary decided whether \((i,j)\)  belongs to \(R\) .
As such, the circuit description of a language \(A\)  is defined to be the set of all circuits that succinctly describe words in \(A\)  [1]}.
For more about circuit description, see [3]}, [1]}, [5]}.
